Gaps between prime numbers

Abstract

ABSTRACT. Let dn = Pn+i ~Pn denote the nth gap in the sequence of primes. We show that for every fixed integer A; and sufficiently large T the set of limit points of the sequence {(dn/logra, ■ • • ,dn+k-i/logn)} in the cube [0, T]k has Lebesgue measure > c(k)Tk, where c(k) is a positive constant depending only on k. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence {dnf log n} has a finite limit point greater than 1.

On Solutions of the Diophantine Equation px + qy = z2

“…In Lemma 2.3, the solutions of (1) have been restricted to all values of p where p < S = 10 5 . Evidently, formulae (5) and (6) are valid for all values of T , and enable us to find all the solutions of (1) up to any value of S where S is as large as we wish.…”

“…These gaps get larger and larger since the density of primes approaches zero in accordance with the prime number theorem. Many articles have been written on this subject, and a very small fraction of them in [4,5] is given here.…”

All the solutions of the Diophantine Equation $p^x + (p+4)^y = z^2$ when p, (p + 4) are Primes and x + y = 2, 3, 4

“…Numerous articles have been written on prime gaps, a very minute fraction of which is brought [3,4] here. In 1849, A. de Polignac conjectured that for every positive integer k, there are infinitely many primes p such that p + 2k is prime too.…”

The Diophantine Equation px + (p+4)y = z2 when p>3, p+4 are Primes is Insolvable in Positive Integers x, y, z

“…A prime gap is the difference between two consecutive primes. Numerous articles have been written on prime gaps, a very minute fraction of which is brought [4,5] here. In 1849, A. de Polignac conjectured that for every positive integer k, there are infinitely many primes p such that p + 2k is prime too.…”

“…When k = 1, the pairs (p, p + 2) are known as Twin Primes. The first four such pairs are: (3,5), (5,7), (11, 13), (17, 19). The Twin Prime conjecture stating that there are infinitely many such pairs remains unproved.…”